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Unconventional Spontaneous Imbibition into Shale Matrix: Theory and a Methodology to Determine Relevant Parameters

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Abstract

Spontaneous imbibition of fracturing fluids into a shale formation has many practical applications for shale gas recovery. Because of the strong solid–liquid interaction in low-permeability media, Darcy law is not always adequate for describing liquid flow process in a shale formation. This unconventional (non-Darcian) flow behavior, however, has not been given enough attention in the shale gas community. The current study develops a systematic methodology to address this important issue. We first review related studies in the literature on relationship between liquid flux and hydraulic (or pressure) gradient in low-permeability media; the unconventional flow behavior is characterized by nonlinearity of the relationship. Then, we propose a phenomenological model for liquid flow in shale (in which liquid flux is a power function of pressure gradient) and develop an analytical solution to a one-dimensional spontaneous imbibition problem that obeys the model. The validity of our model is verified by satisfactory comparisons of theoretical and observed relationships between cumulative imbibition and time. The potential mechanisms of the unconventional flow are discussed. Furthermore, based on the developed analytical solution, we propose a laboratory test methodology to estimate parameters for the phenomenological model from spontaneous imbibition tests.

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Abbreviations

A :

Cross-sectional area of shale column

D :

A variable defined in Eq. 12

g :

Gravitational acceleration

i :

Hydraulic gradient

I :

Threshold gradient

I*:

A parameter defined in Eq. 9a

\(i_{1}\) :

A parameter in Eq. 3a–3c

K :

Hydraulic conductivity

k :

Permeability

k \(^\prime \) :

A parameter in Eq. 3a–3c

\(k_r^*\) :

An analog of relative permeability for liquid flow (Eq. 10)

M :

Cumulative imbibition

N :

A parameter in Eq. 3a–3c

n :

A positive parameter in Eq. 10

\(p_\mathrm{c}\) :

Capillary pressure

q :

Water flux

t :

Time

x :

Location

\(\alpha \) :

A positive constant in Eq. 7

\(\alpha ^{\prime }\) :

A fitting parameter in Eq. 23

\(\beta \) :

A fitting parameter in Eq. 23

\(\lambda \) :

transformation variable defined in Eq. 17

\(\mu \) :

Water viscosity

\(\rho \) :

Water density

\(\theta \) :

Water content

\(\theta _i\) :

Initial water content

\(\theta _0\) :

Water content at x \({=}\) 0

\(\theta ^*\) :

Dimensionless water content (Eq. 24)

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Acknowledgments

The original version of this paper was reviewed by Dr. Dan Georgi from Aramco Research Center (Houston). We also thank the constructive comments from the Associated Editor and the three anonymous reviewers.

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Authors and Affiliations

  1. Aramco Services Company: Aramco Research Center-Houston, 16300 Park Row, Houston, TX, 77084, USA

    Hui-Hai Liu, Bitao Lai & Jinhong Chen

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  1. Hui-Hai Liu
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  2. Bitao Lai
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Correspondence to Hui-Hai Liu.

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Liu, HH., Lai, B. & Chen, J. Unconventional Spontaneous Imbibition into Shale Matrix: Theory and a Methodology to Determine Relevant Parameters. Transp Porous Med 111, 41–57 (2016). https://doi.org/10.1007/s11242-015-0580-z

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